# Strain Energy – Definition, Formula, Derivation

By BYJU'S Exam Prep

Updated on: September 25th, 2023

**Strain Energy** is the energy stored due to the straining of the material. A structure, when subjected to external loads, experiences deformations. To resist these external loads, internal forces are developed. These internal forces are stored in the form of energy in the structure, and it helps the structure regain its original size and shape on removal of external loads, given that the structure’s material is within elastic limit. The ability of a material to store strain energy is called resilience.

For example, when a spring linked to a block on a smooth surface is compressed, strain energy is released. The spring’s strain energy is transferred to the block in kinetic energy when it returns to its original length. Only if the stress is below the elastic limit can the strain energy stored in the material be fully transformed into kinetic energy. We will learn more about strain energy and the formula for strain energy with examples in the upcoming sections.

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## What is Strain Energy?

Strain energy is a sort of potential energy stored in a structural element due to elastic deformation. When a part is deformed from its unstressed state, the external work done on it is changed into (and deemed equal to) the strain energy stored in it. When a beam supported at both ends is subjected to a bending moment caused by suspended loads, the beam is said to be deflected from its unstressed state, and strain energy is stored.

Any object made from a deformable material is subjected to a force changes its shape. This change in shape varies with the properties of the material. For example, on the application of a force to rubber, it elongates to a larger length, while on the application of load on a steel structure, the deformation is not quite visible. For a broad understanding of the strain energy concept, let us first apprehend stress and strain.

## Stress and Strain

Any material will generate equal resistive force when we apply force to it. Stress is the resistive force generated inside the material per unit area. Stress is defined as the ratio of applied force (F) to the area of cross-section (A). Material deforms when force is applied to it. Longitudinal Strain is the linear deformation (length change) per unit length. Longitudinal strain is defined as the ratio of change in length (∆L) to original length (L).

### Relation Between Stress and Strain

According to Hooke’s law, strain is linearly proportional to stress for minor deformations of elastic materials. In an elastic material, Hooke’s law is only true for tiny deformations (up to the proportionate limit).

⟹Stress ∝ Strain

Stress=E x Strain

Where E = Proportionality constant and is called the elastic strain energy formula.

## Stress Vs Strain Graph

The stress vs strain graph for a given material is obtained by applying a gradually increasing force to it and plotting the applied stress with the related strain.

In the above graph:

The proportional limit is at point A.

The elastic limit is at point B.

The yield point is point C.

The letter D stands for ultimate strength.

The fracture point is represented by E.

The elastic region is represented by the area under the curve O to B, while the area under the curve B represents the plastic region to E.

In the region OA, the material obeys Hooke’s law, i.e., stress is directly proportional to strain. When the material is unloaded, it returns to its original dimensions, and all the stored strain energy is released.

## Strain Energy Formula Derivation

When we exert force on a structure, it deforms. The external force will exert force on the structure, which will be stored as strain energy in the material. Strain energy (U) is the energy stored within the recoverable part of the stress-strain curve. The region under the stress-strain curve is known as strain energy. The area above the stress strain is called complementary energy (U*). For a linear elastic system U = U*.

Proof resilience per unit volume is termed the Modulus of resilience. The area of the stress-strain curve gives it within the proportionality limit. Toughness is the ability of a material to store strain energy up to fracture. Modulus of toughness is defined as the toughness per unit volume. The Modulus of toughness gives the area of the stress-strain curve up to fracture.

### Assumptions for Strain Energy Formula

We will need to make some assumptions about the ideal state before deducing the strain energy formula.

- The material is elastic.
- Developed stress is within proportionality limits.
- Gradually applied loading is considered.

Assume a material with original length L and area of cross-section A, deforms by x unit. As per Hooke’s law:

Stress=E∗Strain

⟹F/A=Ex/L

⟹F=EAx/L

Since loading is applied gradually, force F increases gradually along with deformation. Thus, work done can be given as:

The strain energy formula can also be written as:

U=(1/2)x(Stress^{2}/E)×Volume of material

U=(σ^{2}/2E)×V

## Strain Energy per Unit Volume

When the uniform distribution of strain energy takes place, the strain energy stored in a material per unit volume is termed as strain energy density.

u=Total strain energy stored in a body/ Volume of the material= 12×Stress×Strain

u=σ^{2}/2E

## Strain Energy Stored due to Gradually Applied Loading

Strain energy is the area under a load-deformation curve. For gradually applied loading formula for strain energy is given by:

U= area of load Vs deflection curve

U=12×P×∆ = 12×P×PL/AE

⟹U=P^{2}L/2AE

## Strain Energy due to Suddenly Applied Loading

Strain energy is the area under the load-deformation curve. For the gradually applied loading formula for strain, energy is given by:

U= area of load Vs deflection curve

U=1/2xP×∆ =1/2xP×PL/AE

⟹U=P^{2}L/2AE

- The instantaneous stress developed due to suddenly applied load is twice that of the gradually applied load.
- The instantaneous elongation due to a suddenly applied load is twice that of the gradually applied load.
- The instantaneous strain energy stored in the body due to suddenly applied load is four times more than that due to the gradually applied load.

### Strain Energy due to Shear Force

U=∫S_{x}^{2}dx/2A_{r}G

Where, S_{x} = Shear force at any section

A_{r}= Reduced shear area

G = Shear modulus

### Strain Energy due to Bending Moment

U=∫M_{x}^{2}dx/2EI

Where, M_{x} = Bending Moment

I = Moment of inertia about NA

### Strain Energy due to Torque

U=∫T_{x}^{2}dx/2GI_{p}

Where, T_{x} = Applied torque

I_{P} = Polar moment of inertia